Advantages of laying a horse down;
If done properly, it can prove their trust in you.
You can get on a horse without a mounting block or if you're physically disabled.
It's a fun trick.
If desired you can force a horse to stay down & put up with all sorts, in the name of desensitising - called 'flooding'.
It can be dangerous, especially if done badly.
It can have nothing to do with trust and be forced, which can cause a horse to become 'shut down', especially if also exposed to other 'scary' stuff it has no option but to submit to.
If not taught well, horses may do this when not cued.
No, IMO there is no foolproof way that's 'absolutely' safe to teach this, but ensuring you have a sound, trusting relationship with the horse first and using methods to teach that don't involve force are safest.
For trick & fun stuff training generally, "Games" to keep my horse thinking?
is a good thread. ImagineAHorse is a good one to look up for inspiration & ideas. 'Clicker Training' is an effective way to train & the principles behind it are sound & effective no matter what/how you train.